7. Casimir Energies at Finite Temperature

In this section, we consider free fields in finite volumes, maintained in thermal equilibrium at a temperature . Introducing a partition function for the complete system, one obtains as usual

where is the Helmholtz free energy, the total energy and the entropy.

The partition function at finite may be obtained from the path integral expression for the vacuum-to-vacuum amplitude by taking an imaginary time coordinate, in which boson fields have periodicity .

We consider first a scalar field with mass , constrained to have periodicities in space directions. The field is maintained in thermal equilibrium at a temperature , and thus has period in the imaginary time direction. Suppressing an irrelevant normalization factor, the vacuum-to-vacuum amplitude is given by

where the partition function may be written

In the zero temperature limit the summation in (4.3) is replaced by an integral over and the partition function becomes

where an irrelevant additive constant independent of has been dropped. The corresponding energy is then given by , as in Sections 2 and 3 above.

In the case , writing , one may use Eqs. (3.2) and (A.13) to obtain

A convenient representation at low temperatures (high ) is obtained using Eq. (A.13):

The corresponding representation for high temperatures (low ) is obtained from Eq. (A.14) (cf. [47]):

[ Figure 10 ] Energy and entropy as a function of inverse temperature for a massless scalar field in three space dimensions constrained to have unit periods along orthogonal directions.

When , no constraints are imposed, and Eqs. (7.5), (7.6) and (7.7) yield the standard result for a free Bose gas:

Figures 7.1a and b show the internal energy and entropy deduced from Eqs. (7.5), (7.6) and (7.7) with and using relations (7.1). At low temperatures (high ), only the first term of Eq. (7.6) survives. The internal energy tends to the zero temperature Casimir value. The entropy tends to zero, reflecting the uniqueness of the vacuum state (third law of thermodynamics). The second and third terms of Eq. (7.6) represent respectively the contributions of discrete modes in the constrained directions and continuous modes in the unconstrained directions, weighted with the Bose-Einstein distribution at temperature . The modes in the constrained directions have a non-zero minimum energy, and the second term in (7.6) becomes exponentially small when the temperature falls below this minimum energy () in Figs. 7.1a and b. The modes in the unconstrained directions have zero minimum energy, yielding a power law form for the third term. In the high-temperature limit (), only the third term of Eq. (7.7) survives, yielding the Planck result (7.8), independent of .

When , the function in Eq. (7.5) exhibits a pole. The third term in Eq. (7.6) gives the contribution of modes in the unconstrained directions, and is expected to vanish, but instead formally exhibits a pole. The term is seen to vanish if the contributions of low-frequency modes are regularized by a finite before the dimensionality is taken to zero. In this case, the partition function for a free Bose gas is given approximately by

where is the incomplete gamma function. For positive integer (7.9) yields the result (7.8) with corrections of order . For , Eq. (7.8) yields zero in the limit . Thus when , Eq. (7.6) is valid if the last term is set to zero.

In the high-temperature expansion (7.7) the pole at appears in the first term. It may removed in analogy to the low-temperature case above by subtracting the free Bose gas result (7.8) yielding for the first term

where . In the case this becomes

The resulting internal energy and entropy for are given in Fig. 7.1c. In the high-temperature limit, the first correction to the Planck form is .

The results for fields at finite temperature obeying periodicity constraints may be extended to fields with other boundary conditions using the relations (3.8) and (4.6), (4.8) just as in the zero temperature case. (In the analogue of Eq. (3.8), however, the sum over must be extended to include , since for finite when .) With periodic and Neumann boundary conditions, modes with exist, and provide a power correction to the zero temperature Casimir energy at high . With Dirichlet, perfect conductor or bag boundary conditions, no such modes can occur, and the corrections are exponential. In the high-temperature limit, corrections to the Planck form with progressively lower powers of temperature are proportional to the areas of progressively smaller subspaces of the boundary. When , the high-temperature result for a vector field obeying either perfect conductor or bag boundary conditions is

where is a complicated function of the (which could be derived from relations given in [20]).

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